Expert: Bobby Soltani - 8/4/2004

Question
   Hi there Bobby.  How are you doing today?  Hope all is well.  I have this word problem that I can't figure out.  I was wondering and hoping  if you can help me.  Thanks.  Here's the problem:

 A 10-foot ladder is placed against a building so that the distance from the bottom of the ladder to the building is 2 feet less than the distance from the top of the ladder to the ground.  What is the distance from the bottom of the ladder to the building?  

It said I will need a Pythagorean Theorem to solve this problem: a2+b2=c2, “the sum of the legs of a triangle – squared is equal to the hypotenuse squared.”  

Answer
Hi David,

For this problem, try to imagine the ladder forming a triangle with the wall and the ground.  The ladder forms the hypotenuse (c), the ground is one side (a), and the wall is the vertical side (b).  It helps to draw a picture.
From Pythagorean's theorem, we know that a^2+b^2 = c^2.  We also know that c = 10 ft.  We need to find a and b.

The problem says that the distance from the bottom of the ladder to the building, (a), is 2 feet less than the distance from the top to the ground (b).  We can write this as
a = b - 2

Now, we have two equations and two unknowns.  Let's solve them.
1) a = b - 2
2) a^2 + b^2 = 10^2
First, we'll plug (b-2) in for a in the second equation.

(b-2)^2 + b^2 = 100
(b-2)(b-2)+b^2 = 100
b^2 - 4b + 4 + b^2 =100
2b^2 - 4b =96
2b^2 - 4b - 96 = 0
We can use the quadradic equation to get b = 8 or -6.
Obviously, we can discard the -6 because you can't have negative distances.  The questions asks for (a) so we can use the first equation to get it.
a = b - 2
a = 8 - 2
a = 6

The answer is 6 feet from the building.

I hope this helps you out.  Let me know if you have more questions.  Have a great day.

Bobby